A Visual Proof of Girard's Theorem

Girard's theorem states that the area of a spherical triangle is given by the spherical excess: , where the interior angles of the triangle are , , , and the radius of the sphere is 1.

Rewriting the formula in terms of the exterior angles ', ', and ' gives the equivalent formula .

You can use the sliders to change the exterior angles of the (empty) spherical triangle, shifting the area of the two equivalent triangular gaps without changing their area. When you move the "align" slider all the way to the right, the colored spherical lunes align. A spherical lunar-shaped gap is formed with angle , whose area is double the area of the original spherical triangle. On the other hand, the area of a spherical lune is when the radius is 1. Therefore the area of the original spherical triangle is .